Legal Question in Family Law in California

Unless either you or the biological father choose to file for a paternity judgment, there is no reason for you to file for anything. If the public record shows the father as unknown, you get sole custody by default unless and until he steps forward and seeks to be established as the father, or you decide the child needs their father, and/or you need or want the father to pay the child support that the child is entitled to from its father, and so you file a paternity action against the father.

I usually see these circumstances as paternity cases waiting to happen. If father is employed, my guess is that you could use support for the child. If this is the case, filing a paternity petition with a motion for custody and support would be in order.